We present in this paper the characterization of the variational structure behind the discrete equations defining the closest-point projection approximation in elastoplasticity. Rate-independent and viscoplastic formulations are considered in the infinitesimal and the finite deformation range, the later in the context of isotropic finite-strain multiplicative plasticity. Primal variational principles in terms of the stresses and stress-like hardening variables are presented first, followed by the formulation of dual principles incorporating explicitly the plastic multiplier. Augmented Lagrangian extensions are also presented allowing a complete regularization of the problem in the constrained rate-independent limit. The variational structure identified in this paper leads to the proper framework for the development of new improved numerical algorithms for the integration of the local constitutive equations of plasticity as it is undertaken in Part II of this work.